On the local behavior of the Ramanujan $\tau$-function

Let $\tau(n)$ be the Ramanujan function given by \[ q\prod_{k\ge 1} (1-q^k)^{24}=\sum_{n\ge 1} \tau(n) q^n. \] We show that if $k$ is any positive integer such that \begin{equation} (1)\qquad\qquad\tau(j)\ne 0\quad {\text{for any}}\quad j\in \{1,\ldots,k\}, \end{equation} then for every permutation $\sigma$ of $\{1,2,\ldots,k\}$, there exist infinitely many positive integers $n$ such that \[ |\tau(n+\sigma(1))|<|\tau(n+\sigma(2))| < \cdots < |\tau(n+\sigma(k))|. \] Condition (1) holds with $k=982149821766199295999\sim 9\cdot 10^{20}$ and it is conjectured that it holds for all $k$. The proof uses sieves and the truth of the Sato-Tate conjecture for the Ramanujan $\tau$-function. (Joint work with Yuri Bilu.)